Hingeless, large-throw negative stiffness structure

ABSTRACT

A negative stiffness structure for vibration isolation, shock mitigation, and/or signal processing includes a flexible tensile member and a curved compressive member. A first end of the tensile member is attached to a first structure. A first end of the curved compressive member is coupled to a first structure and a second end of the curved compressive member is coupled to a second end of the flexible tensile member. A length of the tensile member is greater than a length of the compressive member. A tip of the negative stiffness structure is configured to exhibit a negative stiffness mechanical response to a load applied to the tip. The negative stiffness mechanical response acts in a direction orthogonal to the length of the tensile member.

FIELD

The following description relates generally to negative stiffnessstructures and, more particularly, hingeless negative stiffnessstructures.

BACKGROUND

A variety of non-linear structures exhibit negative mechanicalstiffness, such as snap-through beams and buckling beams. Negativestiffness may also be exhibited by various combinations and arrangementsof springs and/or beams with pinned or clamped boundaries. For instance,negative stiffness may be exhibited due to over-rotation of one of thecomponents, or rolling or sliding contact between components. Negativestiffness mechanisms are useful in a variety of applications, includingvibration isolation, shock mitigation, and signal processing.

However, related art negative stiffness mechanisms may incorporate pins,hinges, or sliding mechanical joints which increase the complexity andcost of the mechanism. Related art negative stiffness mechanisms alsotypically include a bearing component, which increases the cost of themechanism and increases friction and wear, which may limit the usefullife of the mechanism. Additionally, related art negative stiffnessmechanisms generally have a relatively small range of motion (e.g., alow throw to beam length ratio). For instance, related art negativestiffness mechanisms may be limited to a beam length to throw ratio ofapproximately or about 20:1 or 10:1.

SUMMARY

The present disclosure is directed to various embodiments of negativestiffness structures. In one embodiment, the negative stiffnessstructure includes at least one flexible tensile member and at least onecurved compressive member. A first end of the flexible tensile member iscoupled to a first structure. A first end of first end of the curvedcompressive member is coupled to a second structure and a second end ofthe curved compressive member is coupled to a second end of the flexibletensile member. A length of the flexible tensile member is greater thana length of the curved compressive member. A tip of the negativestiffness structure is configured to exhibit a negative stiffnessmechanical response to a load applied to the tip. The negative stiffnessmechanical response acts in a direction orthogonal to the length of thetensile member.

The compressive member may have a substantially uniform thickness or anon-uniform thickness. The at least one curved compressive member mayinclude a stack of a series of curved compressive members. The at leastone flexible tensile member may include a series of tensile members. Theat least one compressive member may be a rectangular beam. The at leastone tensile member may be a beam, a rope, a cable, a rod, a chain, or abelt.

The negative stiffness structure may also include a second curvedcompressive member having a first end coupled to the second structureand a second end coupled to the second end of the tensile member. Thefirst compressive member may be buckled in a first direction and thesecond compressive member may be buckled in a second direction oppositethe first direction. The first and second compressive members may becosine shaped. The negative stiffness structure may also include firstand second wedge-shaped inserts between the first compressive member andthe second compressive member. The first wedge-shaped insert may beproximate the first ends of the first and second compressive members,and the second wedge-shaped insert may be proximate the second ends ofthe first and second compressive members.

The first structure may include a pair of curved walls defining atapered recess. The first end of the tensile member may be received inthe tapered recess. The negative stiffness structure may include anactuator coupled to the first end of the tensile member. The actuatormay be configured to adjust tension applied to the tensile member oradjust an effective length of the tensile member to vary the mechanicalresponse of the tip. The negative stiffness structure may include aclamping actuator coupled to the first end of the tensile member. Theclamping actuator may be configured to clamp the tensile member at aplurality of positions along a length of the tensile member. The secondstructure may include a clutching mechanism including a first clutchingmember having a convex surface and a second clutching member having aconcave surface mating with the convex surface of the first clutchingmember. The second clutching member may be configured to slide along theconvex surface of the first clutching member between a first angularposition and a second angular position.

In one embodiment, the negative stiffness structure includes at leastone flexible tensile member, an actuator coupled to a first end of theat least one flexible tensile member, and first and second curvedcompressive members. First ends of the curved compressive members arecoupled to a structure. Second ends of the curved compressive membersare coupled to a second end of the flexible tensile member. A length ofthe tensile member is greater than a length of each of the first andsecond curved compressive members. The first curved compressive memberis curved in a first direction and the second curved compressive memberis curved in a second direction opposite the first direction. A tip ofthe negative stiffness structure is configured to exhibit a mechanicalresponse when a load is applied to the tip. The actuator is configuredto vary the mechanical response of the tip. The actuator may be atoothed gear, a pulley, or a clamp configured to clamp the tensilemember at a series of positions along the length of the tensile member.

This summary is provided to introduce a selection of concepts that arefurther described below in the detailed description. This summary is notintended to identify key or essential features of the claimed subjectmatter, nor is it intended to be used in limiting the scope of theclaimed subject matter.

BRIEF DESCRIPTION OF THE DRAWINGS

These and other features and advantages of embodiments of the presentdisclosure will become more apparent by reference to the followingdetailed description when considered in conjunction with the followingdrawings. In the drawings, like reference numerals are used throughoutthe figures to reference like features and components. The figures arenot necessarily drawn to scale.

FIG. 1A is a schematic side view of a negative stiffness structureincluding a tensile member and a pair of compressive members accordingto one embodiment of the present disclosure in a first position;

FIG. 1B is a side view of the negative stiffness structure of FIG. 1A ina second position by tensioning the tensile member;

FIG. 1C is a side view of the negative stiffness structure of FIG. 1A ina third position by deflecting a tip of the negative stiffness structureupward;

FIG. 1D is a side view of the negative stiffness structure of FIG. 1A ina fourth position by deflecting a tip of the negative stiffnessstructure downward;

FIG. 2 is a graph illustrating the mechanical response of a negativestiffness structure according to one embodiment of the presentdisclosure for a range of different effective lengths of the tensilemember;

FIG. 3 is a schematic side view of a negative stiffness structureaccording to another embodiment of the present disclosure;

FIG. 4 is a schematic side view of a negative stiffness structureaccording to another embodiment of the present disclosure;

FIG. 5 is a schematic side view of a negative stiffness structureaccording to another embodiment of the present disclosure;

FIG. 6 is a schematic side view of a negative stiffness structureaccording to another embodiment of the present disclosure;

FIG. 7 is a graph illustrating the mechanical response of the negativestiffness structure of FIG. 6 for a range of different clampingpositions of the tensile member;

FIGS. 8A-8E are schematic side views of a negative stiffness structureaccording to another embodiment of the present disclosure;

FIG. 9 is a schematic side view of a negative stiffness structureaccording to another embodiment of the present disclosure; and

FIG. 10 is a side view of a negative stiffness structure according toone embodiment of the present disclosure incorporated into a hoststructure.

DETAILED DESCRIPTION

The present disclosure is directed to various embodiments of a negativestiffness structure. Embodiments of the negative stiffness structure areconfigured to vary the mechanical response of the structure (i.e., atunable or variable negative stiffness structure). Embodiments of thenegative stiffness structure are also hingeless. The negative stiffnessstructures of the present disclosure may be incorporated into anydesired structure or device depending on the intended purpose orfunction of the negative stiffness structure, such as, for instance, asa tuned-mass damper or as a mechanism for vibration isolation, shockmitigation, or signal processing. For instance, the negative stiffnessstructures of the present disclosure may be incorporated into astructure as an equipment mount (e.g., a mount for a gyroscope), as acomponent of an active or passive vehicle suspension (e.g., forvibration isolation between the engine and the chassis and/or between awheel and the road), or as a component of an aircraft (e.g., forvibration isolation between a helicopter blade and a hub of thehelicopter).

With reference now to FIGS. 1A-1D, a negative stiffness structure 100according to one embodiment of the present disclosure includes a pair ofcompressive elements or members 101, 102 and a tensile element or member103. In the illustrated embodiment, the compressive members 101, 102 areplaced in compression and the tensile member 103 is placed in tension.Such structures in which the components are segregated into pure tensionand pure compression members are known as tensegrity structures. Aninner or base end 104, 105 of each of the compressive members 101, 102,respectively, is fixedly coupled (e.g., clamped or pinned or coupledthrough a flexible member (e.g., a rubber bushing)) to a first structure106 and outer ends 107, 108 of the compressive members 101, 102 are free(i.e., the compressive members 101, 102 are cantilevered from the firststructure 106). In the illustrated embodiment, the inner ends 104, 105and the outer ends 107, 108, respectively, of the compressive members101, 102 are coupled together. Additionally, in the illustratedembodiment, an inner or base end 109 of the tensile member 103 iscoupled to a second structure 110 and an outer end 111 of the tensilemember 103 is connected to the outer ends 107, 108 of the compressivemembers 101, 102. Additionally, the tensile member 103 may pass throughor around the first structure 106 such that the tensile member 103 isnot coupled to the first structure 106. Together, the outer ends 111,107, 108 of the tensile member 103 and the compressive members 101, 102define a tip 112 of the negative stiffness structure 100.

Although in the illustrated embodiment, the negative stiffness structure100 includes a pair of compressive members 101, 102, in one or morealternate embodiments, the negative stiffness structure 100 may have anyother suitable number of compressive members, such as, for instance,from one to twenty. In one embodiment, the negative stiffness structure100 may include one or more stacks of compressive members, such as, forinstance, stacks of up to ten or more compressive members (e.g., upperand/or lower stacks of compressive members). Although in one embodimenteach stack may include the same number of compressive members (e.g., thenumber of compressive members in the upper and lower stacks may besymmetric about the tensile member 103), in one or more alternateembodiments, the stacks may have different numbers of compressivemembers. Additionally, in the illustrated embodiment, the compressivemembers 101, 102 are rectangular beams, although in one or morealternate embodiments, the compressive members 101, 102 may have anyother suitable shape.

Additionally, in the illustrated embodiment, the compressive members101, 102 are preformed into a curved or contoured shape (e.g., a bellcurve or a cosine shape with a single hump). In one embodiment, thecompressive member 101, 102 may have any other suitable shape afterbeing compressed. The compressive members 101, 102 in the illustratedembodiments are also curved or contoured in opposite directions (e.g.,the upper compressive member 101 is curved upward and the lowercompressive member 102 is curved downward). Accordingly, in theillustrated embodiment, the inner and outer ends of the respectivecompressive members contact each other and intermediate portions of thecompressive members 101, 102 between the inner and outer ends are spacedapart, by a maximum amplitude h. For instance, in the illustratedembodiment, the central portions of the compressive members 101, 102 arespaced apart by amplitude h. Additionally, although in the illustratedembodiment each of the compressive members 101, 102 has a substantiallyconstant thickness t, in one or more alternate embodiments, one or moreof the compressive members 101, 102 may have a non-uniform thickness.Furthermore, although in the illustrated embodiment the uppercompressive member 101 has the same or substantially the same thicknessprofile as the lower compressive member 102, in one or more alternateembodiments, the upper compressive member 101 may have a differentthickness profile than the lower compressive member 102.

In one embodiment, the compressive members 101, 102 may be formed fromany material having a relatively high elastic strain limit, such as, forinstance, fiberglass, titanium, or combinations thereof. In one or morealternate embodiments, the compressive members 101, 102 may be formedout of any other suitable material, such as, for instance, steel,silicon, or combinations thereof.

With continued reference to the embodiment illustrated in FIGS. 1A-1D,the tensile member 103 may be any structure suitable for carryingtensile loads, such as, for instance, a beam, a rope (e.g., a wire orKevlar rope), a cable (e.g., a braided cable), a thin rod, a chain, or adrive belt. In one embodiment, the tensile member 103 may be arectangular beam that is the same or similar to one of the compressivemembers 101, 102, but without the preformed shape (e.g., the tensilemember 103 may be a flat rectangular beam). Additionally, in oneembodiment, the tensile member 103 may be flexible such that the tensilemember 103 has a low bending stiffness, but is stiff when a tensile loadis applied to the tensile member 103. Additionally, the tensile member103 may have a low or relatively low bending stiffness in only oneplane, which may aid in restricting the motion of the negative stiffnessstructure 100 to a single plane (e.g., the tensile member 103 may be aroller chain with low bending stiffness in only one plane). In one ormore alternate embodiments, the tensile member 103 may have a bendingstiffness configured to provide a positive stiffness to the negativestiffness structure 100 that would stabilize the negative stiffnessstructure 100 into a finite stiffness structure. If the positivestiffness of the tensile member 103 is only slightly higher than thenegative stiffness of the compressive members 101, 102, the positive andnegative stiffnesses combine to create a quasi-zero-stiffness (QZS)structure 100.

In an initial state illustrated in FIG. 1A, the tensile member 103 hasan initial length L₁ and the linear distance between the inner ends 104,105 and the outer ends 107, 108 of the compressive members 101, 102 hasan initial length L₂. In the illustrated embodiment, the length L₁ ofthe tensile member 103 is longer than the initial length L₂ between theinner ends 104, 105 and the outer ends 107, 108 of the compressivemembers 101, 102. As illustrated in FIG. 1B, the effective length of thetensile member 103 and/or the tension applied to the tensile member 103may be varied based on the desired mechanical response of the negativestiffness structure 100 (i.e., the negative stiffness response of thestructure 100 may be adjusted by changing the tension applied to thetensile member 103 and/or the effective length of the tensile member103). Tensioning the tensile member 103 is configured to increase thecompressive load on the compressive members 101, 102 by decreasing thedistance L₂′ between the inner ends 104, 105 and the outer ends 107, 108of the compressive members 101, 102. Additionally, as illustrated inFIG. 1B, increasing the compressive load on the compressive members 101,102 increases the maximum amplitude h′ between the intermediate portionsof the compressive members 101, 102. Tension may be applied to thetensile member 103 by pulling on the inner end of the tensile member 103with any suitable mechanism. In the embodiment illustrated in FIG. 1B,the tension applied to the tensile member 103 may be increased by movingthe second structure 110 to which the inner end of the tensile member103 is fixedly coupled by a distance e in a direction away from theouter end 111 of the tensile member 103. In another embodiment, theinner end 109 of the tensile member 103 may be coupled to an actuator(e.g., a pulley or a gear) configured to decrease the effective lengthof the tensile member 103 and thereby increase the tension load appliedto the tensile member 103.

As illustrated in FIG. 1C, when an upward force is applied to the tip112 of the negative stiffness structure 100 (i.e., the common outer ends111, 107, 108 of the tensile member 103 and the compressive members 101,102) such that the tip 112 is deflected upward in the +y direction, thestructure 100 is configured to initially exhibit a positive stiffnessresisting the upward deflection. However, as the force and the magnitudeof the upward deflection increase, the structure 100 will reach asnap-through point at which the structure 100 will “snap-through” to astable higher position, shown in FIG. 1C. During snap through, thestructure 100 exhibits negative stiffness. That is, the tip 112 of thestructure 100 exhibits an upward force (i.e., a force in the directionin which the load was applied to the tip 112 of the structure 100).Accordingly, the structure 100 exhibits non-linear stiffness across therange of upward deflections of the tip 112 of the structure 100 (i.e.,the structure 100 exhibits both positive and negative stiffness as thetip 112 of the structure 100 is deflected upward).

Similarly, as illustrated in FIG. 1D, when a downward force is appliedto the tip 112 of the structure 100 such that the tip 112 is deflecteddownward in the −y direction, the structure 100 is initially configuredto exhibit a positive stiffness resisting the downward deflection.However, as the force and the magnitude of the downward deflectionincrease, the structure 100 will reach a snap-through point at which thestructure 100 will snap-through to a stable lower position, shown inFIG. 1D. During snap through, the structure 100 exhibits negativestiffness. That is, the tip 112 of the structure 100 exhibits a downwardforce (i.e., a force in the direction in which the load was applied tothe tip 112 of the structure 100).

The force exhibited at the tip 112 of the structure 100 may betransferred to the system or device into which the negative stiffnessstructure 100 is incorporated by any suitable mechanism, such as, forinstance, a tie rod or other suitable push-pull mechanism.

FIG. 2 is a plot showing the resolved vertical force at the tip 112 ofthe structure 100 when the inner end 109 of the tensile member 103 hadbeen pulled by a range of different distances e (see FIG. 1A) fromapproximately or about 4 mm to approximately or about 6 mm away from theouter end 111 of the tensile member 103. The plot shown in FIG. 2 wasobtained from a negative stiffness structure 100 that included a wirerope tensile member 103 having a diameter of approximately or about 1/16inch and an initial length L₁ of approximately or about 150 mm. Each ofthe upper and lower compressive members 101, 102 were preformed inopposite directions into a cosine shape and had an initial length L₂ ofapproximately or about 100 mm between the inner ends 104, 105 and theouter ends 107, 108 of the compressive members 101, 102. Additionally,each of the compressive members 101, 102 was a rectangular beam having awidth of approximately or about 10 mm and a constant thickness t ofapproximately or about 0.3 mm. In the initial state, the maximumdistance or amplitude h between the intermediate portions of the upperand lower compressive members 101, 102 was approximately or about 10 mm.In particular, FIG. 2 shows the resolved vertical force at the tip 112of the structure 100 when the inner end 109 of the tensile member 103had been pulled by a distance e of approximately or about 4.0 mm,approximately or about 4.5 mm, approximately or about 5.0 mm,approximately or about 5.5 mm, and approximately or about 6.0 mm.

As illustrated in FIG. 2, when the inner end 109 of the tensile member103 had been pulled by a distance e of approximately or about 4.0 mmfrom its initial position, the structure 100 exhibited positivestiffness for tip 112 deflections from approximately or about −0.01 m toapproximately or about +0.01 m and exhibited negative stiffness for tip112 deflections from approximately or about −0.01 m to approximately orabout −0.04 m and from approximately or about +0.01 m to approximatelyor about +0.04 m. Additionally, when the inner end 109 of the tensilemember 103 had been pulled by a distance e of approximately or about 4.0mm, the structure 100 exhibited a maximum positive stiffness force ofapproximately or about 0.1 N at a tip 112 deflection of approximately orabout 0.01 m, and a maximum negative stiffness force of approximately orabout 2 N at a tip 112 deflection of approximately or about 0.04 m.Accordingly, the embodiment of the negative stiffness structure 100tested was configured to exhibit maximum negative stiffness at a throw(i.e., deflection of the tip 112) of approximately or about 40% of thelength L₂ between the inner ends 104, 105 and the outer ends 107, 108 ofthe compressive members 101, 102 (i.e., the structure 100 exhibitedmaximum negative stiffness at a throw of approximately or about 40 mmand the initial length L₂ between the inner ends 104, 105 and the outerends 107, 108 of the compressive members 101, 102 was approximately orabout 100 mm).

When the inner end 109 of the tensile member 103 had been pulled by adistance e of approximately or about 6.0 mm, the structure 100 exhibitedpositive stiffness for tip 112 deflections from approximately or about−0.02 m to approximately or about +0.02 m and exhibited negativestiffness for tip 112 deflections from approximately or about −0.02 m toapproximately or about −0.04 m and from approximately or about +0.02 mto approximately or about +0.04 m. Accordingly, increasing the tensionapplied to the tensile member 103 is configured to increase the range oftip 112 deflections over which the structure 100 is configured toexhibit positive stiffness. Additionally, when the inner end 109 of thetensile member 103 had been pulled by a distance e of approximately orabout 6.0 mm, the structure 100 exhibited a maximum positive stiffnessforce of approximately or about 0.9 N at a tip deflection ofapproximately or about 0.02 m and a maximum negative stiffness force ofapproximately or about 0 N at a tip deflection of approximately or about0.04 m.

With reference now to FIG. 3, a negative stiffness structure 200according to another embodiment of the present disclosure includes apair of upper and lower compressive members 201, 202 and a tensilemember 203. Inner ends 204, 205 of the compressive members 201, 202,respectively, are fixedly coupled to a structure 206 and outer ends 207,208 of the compressive members 201, 202 are free. In the illustratedembodiment, the inner ends 204, 205 and the outer ends 207, 208,respectively, of the compressive members 201, 202 are coupled together.The compressive members 201, 202 are preformed into a curved orcontoured shape (e.g., a cosine shape) extending in opposite directions(e.g., the upper compressive member 201 is curved upward and the lowercompressive member 202 is curved downward). Additionally, in theillustrated embodiment, an inner end 209 of the tensile member 203 isoperatively coupled to an actuator 210 and an outer end 211 of thetensile member 203 is coupled to the outer ends 207, 208 of thecompressive members 201, 202. Together, the outer ends 211, 207, 208 ofthe tensile member 203 and the compressive members 201, 202 define a tip212 of the negative stiffness structure 200. The tensile member 203 andthe compressive members 201, 202 may be the same or similar to thetensile member 103 and the compressive members 101, 102 described abovewith reference to the embodiment illustrated in FIGS. 1A-1D.

The actuator 210 is configured to reduce the effective length of thetensile member 203 and/or increase the tension applied to the tensilemember 203 to modify the mechanical response of the structure 200 when aload is applied to the tip 212 of the structure 200. For instance, inone embodiment, the mechanical response of the structure 200 may vary asshown in FIG. 2 depending on the effective length and/or the tensionload applied to the tensile member 203. The actuator 210 may be anysuitable type or kind of actuator configured to reduce the effectivelength of the tensile member 203 and/or increase the tension applied tothe tensile member 203, such as, for instance, a toothed gear mechanismor a pulley.

With reference now to the embodiment illustrated in FIG. 4, a negativestiffness structure 300 according to another embodiment of the presentdisclosure includes a pair of upper and lower compressive members 301,302 and a tensile member 303. Inner ends 304, 305 of the compressivemembers 301, 302, respectively, are fixedly coupled to a first structure306 and outer ends 307, 308 of the compressive members 301, 302 arefree. In the illustrated embodiment, the inner ends 304, 305 and theouter ends 307, 308, respectively, of the compressive members 301, 302are coupled together. The compressive members 301, 302 are preformedinto a curved or contoured shape (e.g., a cosine shape) extending inopposite directions (e.g., the upper compressive member 301 is curvedupward and the lower compressive member 302 is curved downward).Additionally, in the illustrated embodiment, an inner end 309 of thetensile member 303 is coupled to a second structure 310 and an outer end311 of the tensile member 303 is coupled to the outer ends 307, 308 ofthe compressive members 301, 302. Together, the outer ends 311, 307, 308of the tensile member 303 and the compressive members 301, 302 define atip 312 of the negative stiffness structure 300. In one or morealternate embodiments, the inner end 309 of the tensile member 303 maybe coupled to an actuator (e.g., a pulley or a gear), such as, forinstance, the actuator 210 described above with reference to theembodiment illustrated in FIG. 3, configured to modify the mechanicalresponse of the structure 300 when a load is applied to the tip 312 ofthe structure 300. The tensile member 303 and the compressive members301, 302 may be the same or similar to the tensile member 103 and thecompressive members 101, 102 described above with reference to theembodiment illustrated in FIGS. 1A-1D.

With continued reference to the embodiment illustrated in FIG. 4, thenegative stiffness structure 300 also includes a pair of wedge-shapedinserts 313, 314 disposed between the inner ends 304, 305 and the outerends 307, 308, respectively, of the upper and lower compressive members301, 302 (i.e., the wedge-shaped inserts 313, 314 are located atopposite ends between the compressive members 301, 302). That is, in theillustrated embodiment, the wedge-shaped insert 313 is located proximatethe inner ends 304, 305 of the compressive members 301, 302 and thewedge-shaped insert 314 is located proximate the outer ends 307, 308 ofthe compressive members 301, 302. In one embodiment, the wedge-shapedinserts 313, 314 may be made out of any suitable elastic material, suchas, for instance, natural or synthetic rubber (e.g., a syntheticviscoelastic urethane polymer manufactured by Sorbothane, Inc.). Thewedge-shaped inserts 313, 314 may be coupled to the compressive members301, 302 by any suitable process, such as, for instance, bonding,adhering, or molding. In the illustrated embodiment, the wedge-shapedinserts 313, 314 match or substantially match the shape or contour ofthe inner and outer ends 304, 305, 307, 308 of the compressive members301, 302 when the negative stiffness structure 300 is in a neutralposition. Additionally, in one embodiment, the wedge-shaped inserts 313,314 may be configured elastically deform to match the shape or contourof the inner and outer ends 304, 305, 307, 308 of the compressivemembers 301, 302 as the compressive members 301, 302 buckle in responseto an external load applied to the tip 312 of the structure 300. Thewedge-shaped inserts 313, 314 are configured to provide additionalstability to the negative stiffness structure 300 at high deflections ofthe tip 312 of the structure 300 without requiring additionalpre-shaping of the compressive members 301, 302. Accordingly, thewedge-shaped inserts 313, 314 are configured to increase the negativestiffness or mass efficiency of the structure 300.

With reference now to the embodiment illustrated in FIG. 5, a negativestiffness structure 400 according to another embodiment of the presentdisclosure includes a pair of upper and lower compressive members 401,402 and a tensile member 403. Inner ends 404, 405 of the compressivemembers 401, 402, respectively, are fixedly coupled to a first structure406 and outer ends 407, 408 of the compressive members 401, 402 arefree. In the illustrated embodiment, the inner ends 404, 405 and theouter ends 407, 408, respectively, of the compressive members 401, 402are coupled together. The compressive members 401, 402 are preformedinto a curved or contoured shape (e.g., a cosine shape) extending inopposite directions (e.g., the upper compressive member 401 is curvedupward and the lower compressive member 402 is curved downward).Additionally, in the illustrated embodiment, an inner end 409 of thetensile member 403 is coupled to a second structure 410 and an outer end411 of the tensile member 403 is coupled to the outer ends 407, 408 ofthe compressive members 401, 402. Together, the outer ends 411, 407, 408of the tensile member 403 and the compressive members 401, 402 define atip 412 of the negative stiffness structure 400. The tensile member 403and the compressive members 401, 402 may be the same or similar to thetensile member 103 and the compressive members 101, 102 described abovewith reference to the embodiment illustrated in FIGS. 1A-1D.

Still referring to the embodiment illustrated in FIG. 5, the secondstructure 410 includes a pair of curved walls or surfaces 413, 414diverging apart from each other. Together, the curved walls 413, 414define a tapered or wedge-like recess 415. The recess 415 tapers betweena narrower end 416 and a wider end 417 along a direction extending fromthe inner end 409 to the outer end 411 of the tensile member 403. Theinner end 409 of the tensile member 403 is received in the taperedrecess 415 defined by the pair of curved walls 413, 414 of the secondstructure 410. Accordingly, as the tip 412 of the structure 400 isdeflected either upward or downward, a contact point between the tensilemember 403 and the second structure 410 shifts in a direction toward theouter end 411 of the tensile member 403 (e.g., the contact point betweenthe tensile member 403 and the upper or lower curved wall 413, 414 ofthe second structure 410 continuously changes during the stroke of thetip 412 of the structure 400). Varying the contact point between thetensile member 403 and the second structure 410 is configured topassively change the effective length of the tensile member 403 andthereby alter the mechanical response of the structure 400. As usedherein, the “effective length” of the tensile member 403 is defined asthe length of the tensile member 403 from the contact point between thetensile member 403 and one of the curved walls 413, 414 and the outerend 411 of the tensile member 403. In one or more alternate embodiments,the structure 400 may include a plurality of discrete stops configuredto passively adjust the effective length of the tensile member 403. Forinstance, in one embodiment, the curved walls 413, 414 of the secondstructure 410 may include a plurality of discrete projections or edges(e.g., the curved walls 413, 414 may not be smooth). The contact betweenthe discrete projections and the tensile member 403 is configured toadjust the effective length of the tensile member 403.

Additionally, in the embodiment illustrated in FIG. 5, the negativestiffness structure 400 also includes a pair of wedge-shaped inserts418, 419 disposed between opposite ends of the compressive members 401,402. In one embodiment, the wedge-shaped inserts 418, 419 may be thesame or similar to the wedge-shaped inserts 313, 314 described abovewith reference to the embodiment illustrated in FIG. 4. In one or morealternate embodiments, the negative stiffness structure 400 may beprovided without the wedge-shaped inserts 418, 419.

With reference now to the embodiment illustrated in FIG. 6, a negativestiffness structure 500 according to another embodiment of the presentdisclosure includes a pair of upper and lower compressive members 501,502 and a tensile member 503. Inner ends 504, 505 of the compressivemembers 501, 502, respectively, are fixedly coupled to a first structure506 and outer ends 507, 508 of the compressive members 501, 502 arefree. In the illustrated embodiment, the inner ends 504, 505 and theouter ends 507, 508, respectively, of the compressive members 501, 502are coupled together. The compressive members 501, 502 are preformedinto a curved or contoured shape (e.g., a cosine shape) extending inopposite directions (e.g., the upper compressive member 501 is curvedupward and the lower compressive member 502 is curved downward).Additionally, in the illustrated embodiment, an inner end 509 of thetensile member 503 is coupled to a second structure 510 and an outer end511 of the tensile member 503 is coupled to the outer ends 507, 508 ofthe compressive members 501, 502. Together, the outer ends 511, 507, 508of the tensile member 503 and the compressive members 501, 502 define atip 512 of the negative stiffness structure 500. The tensile member 503and the compressive members 501, 502 may be the same or similar to thetensile member 103 and the compressive members 101, 102 described abovewith reference to the embodiment illustrated in FIGS. 1A-1D.

With continued reference to the embodiment illustrated in FIG. 6, thenegative stiffness structure 500 also includes a clamp 513 coupled tothe second structure 510. The clamp 513 is configured to clamp onto thetensile member 503 at various points or positions along the length ofthe tensile member 503. Clamping onto the tensile member 503 isconfigured to actively alter the effective length of the tensile member503 and thereby vary the mechanical response of the structure 500, asdescribed below with reference to FIG. 7. When the clamp 513 engages thetensile member 503, the effective length of the tensile member 503 isdefined as the length of the tensile member 503 from the clamping pointto the outer end 511 of the tensile member 503.

Additionally, although in the illustrated embodiment the negativestiffness structure 500 also includes a pair of wedge-shaped inserts514, 515 disposed between opposite ends of the compressive members 501,502, in one or more alternate embodiments, the negative stiffnessstructure 500 may be provided without the wedge-shaped inserts 514, 515.In one embodiment, the wedge-shaped inserts 514, 515 may be the same orsimilar to the wedge-shaped inserts 313, 314 described above withreference to the embodiment illustrated in FIG. 4.

FIG. 7 is a plot showing the mechanical response of the embodiment ofthe negative stiffness structure 500 illustrated in FIG. 6 for a rangeof different clamping positions at which the clamp 513 is clamped ontothe tensile member 503. As described above, the clamp 513 is configuredto reduce the effective length of the tensile member 503 by clampingonto the tensile member 503. In particular, FIG. 7 illustrates themechanical response of the negative stiffness structure 500 when theeffective length of the tensile member 503 has been reduced from aninitial length L of approximately or about 150 mm to approximately orabout 140 mm, approximately or about 130 mm, and approximately or about120 mm.

When the clamp 513 was not actuated, such that the tensile member 503had an effective length equal to its overall length L of approximatelyor about 150 mm, the structure 500 exhibited positive stiffness from arange of tip 512 deflections from approximately or about −0.02 m toapproximately or about +0.02 m and the structure 500 exhibited negativestiffness for a range of tip 512 deflections from approximately or about−0.02 m to approximately or about −0.04 m and from approximately orabout +0.02 m to approximately or about +0.04 m. Additionally, thestructure 500 exhibited a maximum positive stiffness force ofapproximately or about 0.9N and a maximum negative stiffness force ofapproximately or about 0N. When the clamp 513 was actuated such that thetensile member 503 has an effective length of approximately or about 140mm (i.e., the clamp 513 was clamped onto the tensile member 503 atapproximately or about 10 mm from the inner end 509 of the tensilemember 503), the structure 500 exhibited positive stiffness for a rangeof tip 512 deflections from approximately or about −0.02 m toapproximately or about +0.02 m, a maximum positive stiffness force ofapproximately or about 0.5N, and a maximum negative stiffness force ofapproximately or about 0.2N. When the clamp 513 was actuated such thatthe tensile member 503 had an effective length of approximately or about130 mm (i.e., the clamp 513 was clamped onto the tensile member 503 atapproximately or about 20 mm from the inner end 509 of the tensilemember 503), the structure 500 exhibited positive stiffness for a rangeof tip 512 deflections from approximately or about −0.02 m toapproximately or about +0.02 m, a maximum positive stiffness force ofapproximately or about 0.1N, and a maximum negative stiffness force ofapproximately or about 0.7N. When the clamp 513 was actuated such thatthe tensile member 503 had an effective length of approximately or about120 mm (i.e., the clamp 513 was clamped onto the tensile member 503 atapproximately or about 30 mm from the inner end 509 of the tensilemember 503), the structure 500 exhibited negative stiffness across theentire range of tip 512 deflections from approximately or about −0.04 mto approximately or about +0.04 m and exhibited a maximum negativestiffness force of approximately or about 1.2N.

With reference now to the embodiment illustrated in FIGS. 8A-8D, anegative stiffness structure 600 according to another embodiment of thepresent disclosure includes a pair of upper and lower compressivemembers 601, 602 and a tensile member 603. Inner ends 604, 605 of thecompressive members 601, 602, respectively, are fixedly coupled to afirst structure 606 and outer ends 607, 608 of the compressive members601, 602 are free. In the illustrated embodiment, the inner ends 604,605 and the outer ends 607, 608, respectively, of the compressivemembers 601, 602 are coupled together. The compressive members 601, 602are preformed into a curved or contoured shape (e.g., a cosine shape)extending in opposite directions (e.g., the upper compressive member 601is curved upward and the lower compressive member 602 is curveddownward). Additionally, in the illustrated embodiment, an inner end 609of the tensile member 603 is coupled to a second structure 610 and anouter end 611 of the tensile member 603 is coupled to the outer ends607, 608 of the compressive members 601, 602. Together, the outer ends611, 607, 608 of the tensile member 603 and the compressive members 601,602 define a tip 612 of the negative stiffness structure 600. Thetensile member 603 and the compressive members 601, 602 may be the sameor similar to the tensile member 103 and the compressive members 101,102 described above with reference to the embodiment illustrated inFIGS. 1A-1D.

With continued reference to the embodiment illustrated in FIGS. 8A-8D,the negative stiffness structure 600 also includes a clutching orbraking mechanism 613 coupled to the first structure 606. As describedin more detail below, the clutching mechanism 613 is configured toenable the angular position of the tensile member 603 and the upper andlower compressive members 601, 602 to be adjusted while maintaining thesymmetry of the compressive members 601, 602 about the tensile member603. The angular position of the tensile member 603 and the compressivemembers 601, 602 may be adjusted based on the structure or device intowhich the negative stiffness structure 600 is intended to beincorporated and/or based on the nature of the external load applied tothe tip 612 of the negative stiffness structure 600. The clutchingmechanism 613 may be any suitable type of mechanism for reversablylocking and unlocking two surfaces together, such as, for instance, afriction type or kind clutching mechanism or a mechanical locking typeor kind clutching mechanism.

Stiff referring to the embodiment illustrated in FIGS. 8A-8D, theclutching mechanism 613 includes an inner clutching member 614 having anouter interface surface 615 and an outer clutching member 616 having aninner interface surface 617 that matches or substantially matches theshape or contour of the outer interface surface 615 of the innerclutching member 614 (e.g., the interface surfaces 615, 617 of the innerand outer clutching members 614, 616 are complementary). In theillustrated embodiment, the outer interface surface 615 of the innerclutching member 614 is a convex surface and the inner interface surface617 of the outer clutching member 616 is a concave surface that conformsor substantially conforms to the convex outer interface surface 615 ofthe inner clutching member 614. Additionally, in one embodiment, theouter interface surface 615 of the inner clutching member 614 may definean arc centered about the inner end 609 of the tensile member 603. Inone or more alternate embodiments, the interface surfaces 615, 617 ofthe inner and outer clutching members 614, 616 may have any other shapessuitable for enabling the outer clutching member 616 to rotate or pivot(arrow 618) relative to the inner clutching member 614. For instance, inone embodiment, the inner clutching member 614 may include a concaveinterface surface 615 and the outer clutching member 616 may include aconvex interface surface 617 (e.g., the inner and outer clutchingmembers 614, 616 may function similar to a ball and socket joint).Additionally, in the illustrated embodiment, the inner ends 604, 605 ofthe compressive members 601, 602 are coupled to the outer clutchingmember 616.

As illustrated in FIG. 8B, before the angular position of thecompressive members 601, 602 and the outer clutching member 616 havebeen adjusted, the negative stiffness structure 600 is configured tofunction in the same or similar manner as the embodiment described abovewith reference to FIGS. 1A-1D in response to a load applied to the tip612 of the structure 600.

The angular position of the tensile and compressive members 601, 602 maybe adjusted (arrow 618) by reducing the tension applied to the tensilemember 603 or increasing the effective length of the tensile member 603.In one embodiment, the inner end 609 of the tensile member 603 may becoupled to any suitable mechanism configured to adjust the tension ofthe tensile member 603 or adjust the effective length of the tensilemember 603. As illustrated in FIG. 8C, the tension applied to thetensile member 603 may be reduced by moving (arrow 619) the secondstructure 610 to which the inner end 609 of the tensile member 603 isfixedly coupled in a direction toward the outer end 611 of the tensilemember 603. In another embodiment, the inner end 609 of the tensilemember 603 may be coupled to an actuator (e.g., a pulley or a gear)configured to increase the effective length of the tensile member 603and thereby decrease the tension applied to the tensile member 603.

Reducing the tension applied to the tensile member 603 or increasing theeffective length of the tensile member 603 is configured to reduce thecompressive force applied to the compressive members 601, 602 andthereby reduce the friction between the interface surfaces 615, 617 ofthe inner and outer clutching members 614, 616. As illustrated in FIG.8D, the reduced friction between the interface surfaces 615, 617 of theinner and outer clutching members 614, 616 causes or enables the outerclutching member 616 to rotate, pivot, or slide downward (arrow 618)along the interface surface 615 of the inner clutching member 614. Oncethe outer clutching member 616 and the compressive members 601, 602coupled thereto have rotated down (arrow 618) into the desired angularposition, as illustrated in FIG. 8E, the desired angular position may beset by increasing the tension applied to the tensile member 603 ordecreasing the effective length of the tensile member 603 to increasethe compressive force applied to the compressive members 601, 602. Asillustrated in FIG. 8E, the tensile applied to the tensile member 603may be increased or the effective length of the tensile member 603 maybe decreased by moving (arrow 620) the second structure 610 in adirection away from the outer end 611 of the tensile member 603 or byactuating the actuator coupled to the inner end 609 of the tensilemember 603.

The increased compressive force applied to the compressive members 601,602 is configured to draw the outer clutching member 616 into tighterengagement with the inner clutching member 614 and thereby increase thefriction between the interface surfaces 615, 617 of the inner and outerclutching members 614, 616. The increased friction between the interfacesurfaces 615, 617 of the inner and outer clutching members 614, 616 isconfigured to prevent the outer clutching member 616 and the compressivemembers 601, 602 coupled thereto from inadvertently rotating (arrow 618)relative to the inner clutching member 614. In one embodiment, theclutching mechanism 613 is configured to adjust the angular position ofthe tensile member 603 and the compressive members 601, 602 byapproximately or about +/−10 degrees from the neutral position shown inFIG. 8A, although in one or more alternate embodiments, the clutchingmechanism 613 may be configured to adjust the angular position of thetensile member 603 and the compressive members 601, 602 to any othersuitable extent.

Once the negative stiffness structure 600 has been positioned into thedesired angular position, the negative stiffness structure 600 isconfigured to function in the same or similar manner as the negativestiffness structure 100 described above in detail with reference toFIGS. 1A-1D and FIG. 2 when a load is applied to the tip 612 of thestructure 600.

Additionally, in one or more alternate embodiments, the negativestiffness structure 600 may include one or more wedge-shaped insertsdisposed between the compressive members 601, 602. The wedge-shapedinserts may be the same or similar to the wedge-shaped inserts 313, 314described above with reference to the embodiment illustrated in FIG. 4.

With reference now to FIG. 9, a negative stiffness structure 700according to another embodiment of the present disclosure includes apair of upper and lower compressive members 701, 702 and a tensilemember 703. The negative stiffness structure 700 also includes aclutching mechanism 704 that includes an inner clutching member 705 andan outer clutching member 706 configured to selectively pivot or rotate(arrow 707) along the inner clutching member 705 from a first angularposition (shown in solid lines) to a second angular position (shown indashed lines). As described above with reference to the embodimentillustrated in FIGS. 8A-8D, the clutching mechanism 704 is configured toenable the angular position of the tensile member 703 and the upper andlower compressive members 701, 702 to be adjusted, such as, forinstance, based on the structure or device into which the negativestiffness structure 700 is intended to be incorporated and/or based onthe nature of the external load applied to the negative stiffnessstructure 700. Inner ends 708, 709 of the compressive members 701, 702,respectively, are fixedly coupled to the outer clutching member 706 andouter ends 710, 711 of the compressive members 701, 702 are free. In theillustrated embodiment, the inner ends 708, 709 and the outer ends 710,711, respectively, of the compressive members 701, 702 are coupledtogether. The compressive members 701, 702 are preformed into a curvedor contoured shape (e.g., a cosine shape) extending in oppositedirections (e.g., the upper compressive member 701 is curved upward andthe lower compressive member 602 is curved downward). Additionally, inthe illustrated embodiment, an inner end 712 of the tensile member 703is coupled to a structure 713 and an outer end 714 of the tensile member703 is coupled to the outer ends 710, 711 of the compressive members701, 702. Together, the outer ends 714, 710, 711 of the tensile member703 and the compressive members 701, 702 define a tip 715 of thenegative stiffness structure 700. The tensile member 703 and thecompressive members 701, 702 may be the same or similar to the tensilemember 103 and the compressive members 101, 102 described above withreference to the embodiment illustrated in FIGS. 1A-1D. The clutchingmechanism 704 may be the same or similar to the clutching mechanism 613described above with reference to FIGS. 8A-8D. In one embodiment, thestructure 713 also includes an actuator 716 configured to adjust thetension applied to the tensile member 703. Adjusting the tension of thetensile member 703 is configured to increase or decrease the frictionbetween the inner and outer clutching members 705, 706 and thereby lockthe angular position of the compressive members 701, 702 and the tensilemember 703 or permit the outer clutching member 706 and the compressivemembers 701, 702 to rotate (arrow 707) along the inner clutching member705 into a different angular position (shown in dashed lines).

FIG. 10 illustrates a negative stiffness structure 800 according to oneembodiment of the present disclosure integrated with a host structure801. The negative stiffness structure 800 may be the same or similar toany one of the embodiments described above. In the illustratedembodiment, the negative stiffness structure 800 includes a pair ofupper and lower compressive members 802, 803 and a pair of tensilemembers 804, 805. Inner ends 806, 807 of the compressive members 802,803, respectively, are fixedly coupled to a first structure 808. In theillustrated embodiment, the inner ends 806, 807 and outer ends 809, 810,respectively, of the compressive members 802, 803 are coupled together.The compressive members 802, 803 are preformed into a curved orcontoured shape (e.g., a cosine shape) extending in opposite directions(e.g., the upper compressive member 802 is curved upward and the lowercompressive member 803 is curved downward). Additionally, in theillustrated embodiment, inner ends 811, 812 of the tensile members 804,805 are coupled to a second structure 813 and outer ends 814, 815 of thetensile members 804, 805 are coupled to the outer ends 809, 810 of thecompressive members 802, 803. Together, the outer ends 814, 815, 809,810 of the tensile members 804, 805 and the compressive members 802, 803define a tip 816 of the negative stiffness structure 800. The tensilemembers 804, 805 and the compressive members 802, 803 may be the same orsimilar to the tensile member 103 and the compressive members 101, 102described above with reference to the embodiment illustrated in FIGS.1A-1D.

Still referring to the embodiment illustrated in FIG. 10, the hoststructure 801 includes a tie rod 817, a positive stiffness swing arm818, and a spring 819. In the illustrated embodiment, a lower end 820 ofthe tie rod 817 is coupled to the second structure 813 to which theinner ends 811, 812 of the tensile members 804, 805 are coupled. The tierod 817 extends upward from the second structure 813. The swing arm 818includes a horizontal member 821 and a vertical member 822 hingedlycoupled to the horizontal member 821. An inner end 823 of the horizontalmember 821 is coupled to and extends outward from an upper end 824 ofthe tie rod 817. An upper end 825 of the vertical member 822 is hingedlycoupled to an outer end 826 of the horizontal member 821. A lower end827 of the vertical member 822 is coupled to the outer ends 814, 815,809, 810 of the tensile members 804, 805 and the compressive members802, 803. In the illustrated embodiment, the spring 819 extends betweenan intermediate portion of the horizontal member 821 and the secondstructure 813 to which the inner ends 811, 812 of the tensile members804, 805 are coupled.

The mechanical response of the tip 816 of the negative stiffnessstructure 800 is transmitted to the vertical member 822 of the swing arm818. Additionally, the mechanical response of the negative stiffnessstructure 800 may be varied by adjusting the tension applied to thetensile members 804, 805 or adjusting the effective length of thetensile members 804, 805. The negative stiffness structure 800 mayinclude any suitable mechanism for adjusting the tension applied to thetensile members 804, 805 and/or adjusting the effective length of thetensile members 804, 805, such as, for instance, an actuator (e.g., apulley or a gear) coupled to the inner ends 811, 812 of the tensilemembers 804, 805. For instance, in one embodiment, the mechanicalresponse of the negative stiffness structure 800 may be varied as shownin FIG. 2.

While this invention has been described in detail with particularreferences to exemplary embodiments thereof, the exemplary embodimentsdescribed herein are not intended to be exhaustive or to limit the scopeof the invention to the exact forms disclosed. Persons skilled in theart and technology to which this invention pertains will appreciate thatalterations and changes in the described structures and methods ofassembly and operation can be practiced without meaningfully departingfrom the principles, spirit, and scope of this invention, as set forthin the following claims, and equivalents thereof. Although relativeterms such as “outer,” “inner,” “upper,” “lower,” and similar terms havebeen used herein to describe a spatial relationship of one element toanother, it is understood that these terms are intended to encompassdifferent orientations of the various elements and components of theinvention in addition to the orientation depicted in the figures.Additionally, as used herein, the term “substantially,” “about,” andsimilar terms are used as terms of approximation and not as terms ofdegree, and are intended to account for the inherent deviations inmeasured or calculated values that would be recognized by those ofordinary skill in the art. Moreover, when a component is component isreferred to as being “coupled” to another component, it can be directlyattached to the other component or intervening components may be presenttherebetween.

What is claimed is:
 1. A negative stiffness structure, comprising: atleast one flexible tensile member having a first end and a second end,the first end coupled to a first structure; and a first curvedcompressive member having a first end and a second end, the first endcoupled to a second structure, the second end of the first curvedcompressive member directly coupled to the second end of the tensilemember, wherein a length of the tensile member is greater than a lengthof the first curved compressive member, and a tip of the negativestiffness structure is configured to exhibit a negative stiffnessmechanical response to a load applied to the tip, the negative stiffnessmechanical response acting in a direction orthogonal to the length ofthe tensile member.
 2. The negative stiffness structure of claim 1,further comprising a second curved compressive member having a first endcoupled to the second structure and a second end coupled to the secondend of the tensile member.
 3. The negative stiffness structure of claim2, wherein the first curved compressive member is buckled in a firstdirection, and wherein the second curved compressive member is buckledin a second direction opposite the first direction.
 4. A negativestiffness structure, comprising: at least one flexible tensile memberhaving a first end and a second end, the first end coupled to a firststructure; a first curved compressive member having a first end and asecond end, the first end coupled to a second structure, the second endcoupled to the second end of the tensile member; and a second curvedcompressive member having a first end coupled to the second structureand a second end coupled to the second end of the tensile member,wherein: a length of the tensile member is greater than a length of thefirst curved compressive member, a tip of the negative stiffnessstructure is configured to exhibit a negative stiffness mechanicalresponse to a load applied to the tip, the negative stiffness mechanicalresponse acting in a direction orthogonal to the length of the tensilemember, and the first and second curved compressive members are cosineshaped.
 5. The negative stiffness structure of claim 2, furthercomprising first and second wedge-shaped inserts between the firstcurved compressive member and the second curved compressive member, thefirst wedge-shaped insert proximate the first ends of the first andsecond curved compressive members, and the second wedge-shaped insertproximate the second ends of the first and second curved compressivemembers.
 6. The negative stiffness structure of claim 1, wherein thefirst structure comprises a pair of curved walls defining a taperedrecess, and wherein the first end of the tensile member is received inthe tapered recess.
 7. The negative stiffness structure of claim 1,further comprising an actuator coupled to the first end of the tensilemember, the actuator configured to adjust tension applied to the tensilemember or adjust an effective length of the tensile member to vary themechanical response of the tip.
 8. The negative stiffness structure ofclaim 1, further comprising a clamping actuator coupled to the first endof the tensile member, the clamping actuator configured to clamp thetensile member at a plurality of positions along a length of the tensilemember.
 9. The negative stiffness structure of claim 1, wherein thesecond structure comprises a clutching mechanism, the clutchingmechanism comprising: a first clutching member having a convex surface;and a second clutching member having a concave surface mating with theconvex surface of the first clutching member, wherein the secondclutching member is configured to slide along the convex surface of thefirst clutching member between a first angular position and a secondangular position.
 10. The negative stiffness structure of claim 1,wherein the first curved compressive member has a substantially uniformthickness.
 11. A negative stiffness structure, comprising: at least oneflexible tensile member having a first end and a second end, the firstend coupled to a first structure; and a first curved compressive memberhaving a first end and a second end, the first end coupled to a secondstructure, the second end coupled to the second end of the tensilemember, wherein: a length of the tensile member is greater than a lengthof the first curved compressive member, a tip of the negative stiffnessstructure is configured to exhibit a negative stiffness mechanicalresponse to a load applied to the tip, the negative stiffness mechanicalresponse acting in a direction orthogonal to the length of the tensilemember, and wherein the first curved compressive member has anon-uniform thickness.
 12. A negative stiffness structure, comprising:at least one flexible tensile member having a first end and a secondend, the first end coupled to a first structure; and a first curvedcompressive member having a first end and a second end, the first endcoupled to a second structure, the second end coupled to the second endof the tensile member, wherein: a length of the tensile member isgreater than a length of the first curved compressive member, a tip ofthe negative stiffness structure is configured to exhibit a negativestiffness mechanical response to a load applied to the tip, the negativestiffness mechanical response acting in a direction orthogonal to thelength of the tensile member, and the first curved compressive membercomprises a stack of a plurality of curved compressive members.
 13. Thenegative stiffness structure of claim 1, wherein the at least oneflexible tensile member comprises a plurality of tensile members. 14.The negative stiffness structure of claim 1, wherein the compressivemember is a rectangular beam.
 15. The negative stiffness structure ofclaim 1, wherein the at least one tensile member is selected from thegroup consisting of a beam, a rope, a cable, a rod, a chain, or a belt.